I definitely agree with you that
Paige is not related to
Pæga because of the sound difference. Pagan and Payne are the English surnames derived from the medieval given name based on Latin Paganus.
However, you shouldn't use the "gender mismatch" as part of the argument. Like most given names derived from surnames,
Paige was originally given almost exclusively to males. For example, all 16 examples of a person with
Paige as a given name in the 1860 United States census are male. The use of
Paige as a female name is a more recent phenomenon.